# Inelastic Collisions - conceptual trouble

Hi there I'm having a bit of a conceptual difficulty regarding the equations for inelastic collisions. Suppose a body of mass M1, moving at initial velocity V1, collides and sticks to another body, mass M2, moving at some other initial velocity V2. They then move together with a final velocity of V3.

Now I understand that in this situation, kinetic energy is not conserved, though momentum (of the two-particle system) is. I also know that the "missing" kinetic energy must have gone into the production of sound waves, or thermal energy, changes of potential energy (if the bodies deform?) etc. If only because this is what I am taught.

If we suppose that V3 is the only unknown in the above problem, then I can quite easily calculate the magnitude of V3 from the other data (suppose that the collision is linear, to simplify things). Then it is also easy to calculate the final kinetic energy of the two-body system. I've done these problems countless times so no difficulty in the calculatiions.

But what I can't understand: how is it that these simple conservation-of-linear-momentum equations "know" that kinetic energy is not conserved? Surely there would be a hypothetical magnitude for V3 such that kinetic energy would (hypothetically) be conserved -- how do these equations correctly produce a a V3 such that kinetic energy is NOT conserved? In fact, working symbolically with the conservation of momentum equations, I can sort of see why the system's kinetic energy would be decreased, in such a collision. Still, conceptually, I am not happy about how this all gets built into these equations. Last edited:

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Ok, I've looked at the math there and I've done all those calculations before. Conceptually, this is how I see it: linear momentum must be conserved; thus one uses the equations for conservation of linear momentum to express final velocity i.t.o masses and initial velocities. The easily appreciated consequence is that kinetic energy will be lost.

Why, though, does momentum conservation get precedence over kinetic energy conservation? Is it by virtue of the definition of an inelastic collision, that the system's internal energy should change? (Now that I think about it...that actually seems glaringly obvious!)

Still, definitions aside, why is not possible (even in some idealised situation?) to have a sticking-collision whilst conserving kinetic energy?

What else can you say when you measure the the final kinetic energy and find it to be less than the initial in all such cases?

Why, though, does momentum conservation get precedence over kinetic energy conservation?
The Lagrangian is invariant under translations, momentum is a conserved quantity of the symmetry group of translations. Apart that I don't know 'why'.

@neutrino:

If the system (of the two particles) is isolated, and the kinetic energy decreases, then the internal energy of the particles must increase. Since total energy is conserved in the isolated system.

If the final kinetic energy is less than the initial kinetic energy, then the mag. of the final velocity must be less than the mag. of the larger of the two initial velocities (I think...tho I'd better check that algebraically...).
Basically m1(v1)^2 + m2(v2)^2 > m1(v3)^2 + m2(v3)^2...

AHA, so I think I see where this is going...so qualitatively then, because the bodies stick together, at least one component of a body's velocity must decrease?

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